What is the purpose of this popup? I can't imagine a good reason to do it, but if you explain what you're trying to achieve you will get more useful answers.
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JohnGB♦Nov 2 '11 at 23:41

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Given the size of mobile screens, I'm finding complex pop-up modals are usually a bit pointless, as they tend to be as big as the screen anyways. As such, I'd usually suggest clicking Books would cause a full page to slide up over over to show the same information. (One exception being alerts--which still have a use in a pop-up modal UI)
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DA01Nov 3 '11 at 0:58

Part of the argument against having the popover the way you mention is the potential for confusing hierarchy.

Once you've jumped to, say, non-fiction, I'm guessing you want to support the user viewing/filtering by the different categories, jumping from non-fiction to, say, fiction.

At that point, if those options are accessed through the bottom menu, I think you're working against general convention, where filters are above the content they filter (ex. iPad iTunes App).

One could argue that iOS's bottom tab bar sets a new convention, but I think that can be attributed as an outgrowth of the iPhone's patterns, and not a proposal for a more extended inversion of page hierarchy. I think some of the bottom aligned interactions work well with touch interactions, but it's a balancing act with hierarchy concepts.